Super-maximal representations from fundamental groups of punctured surfaces to $\mathrm{PSL}(2,\mathbb{R})$

TL;DR

This paper investigates super-maximal representations of punctured sphere groups into PSL(2,R), revealing their geometric properties, compactness, and symplectic structure, and establishing their uniqueness among components of the relative character variety.

Contribution

It introduces super-maximal representations, proves their geometric and symplectic properties, and characterizes their uniqueness as compact components in the relative character variety.

Findings

Super-maximal representations are totally non hyperbolic.

They are geometrizable via unique holomorphic equivariant maps.

Components of super-maximal representations are compact and symplectomorphic to complex projective space.

Abstract

We study a particular class of representations from the fundamental groups of punctured spheres $\Sigma_{0,n}$ to the group $\text{PSL} (2,\mathbb R)$ (and their moduli spaces), that we call \emph{super-maximal}. Super-maximal representations are shown to be \emph{totally non hyperbolic}, in the sense that every simple closed curve is mapped to a non hyperbolic element. They are also shown to be \emph{geometrizable} (appart from the reducible super-maximal ones) in the following very strong sense : for any element of the Teichm\"uller space $\mathcal T_{0,n}$, there is a unique holomorphic equivariant map with values in the lower half-plane $\mathbb H^-$. In the relative character variety, the components of super-maximal representations are shown to be compact, and symplectomorphic (with respect to the Atiyah-Bott-Goldman symplectic structure) to the complex projective space of dimension $n-3$ equipped with a certain multiple of the Fubiny-Study form that we compute explicitly (this generalizes a result of Benedetto--Goldman for the sphere minus four points). Those are the unique compact components in relative character varieties of $\text{PSL}(2,\mathbb R)$. This latter fact will be proved in a companion paper.